Introduction
So far, we have dealt with situations where everyone who is assigned
to treatment actually receives the treatment.
But this may not always hold:
- What if people don’t read negative posts on their social media?
- What if people don’t pay attention to fact checkers?
- Experiments provide lottery winners with vouchers that can be used
to defray tuition at private schools, but only some of the lottery
winners actually use the vouchers.
- Canvassers are assigned to go door-to-door and encourage people to
vote, but some people are not home.
Further, the people who get treated may be systematically different
from the people who are assigned to the treatment group, but somehow
don’t get treated.
What types of causal effects can we estimate in such situations?
A Motivating Example
Suppose that you are interested in assessing the effect of a
face-to-face canvassing effort on voter turnout. Here’s the design:

Suppose you were the principal investigator of this study. How should
the data from these three groups be analyzed? Which groups (boxes) would
you compare?
Some Terminology
- Compliers: are treated if assigned to the treatment
group, and untreated if assigned to the control group
- Never-takers: are untreated, even if assigned to
the treatment group
- Always-takers: are treated, even if assigned to the
control group (ruled out by design in our specific example)
- Defiers: are treated if assigned to the control
group, and untreated if assigned to the treatment group
(assumed not to exist. This is called the
“monotonicity” or “no-defiers” assumption).
Importantly, these designations define how treatment
status (D) responds to treatment assignment
(Z). They are not defined with respect to potential outcomes (Y).
A graphical illustration

The total shaded area in Panel A (including the cross-hatched
region) represents the average outcome for the subject pool if all
subjects were assigned to the treatment group.
The shaded area in Panel B represents the average outcome if all
subjects were assigned to the control group.
The difference between the shaded areas in Panels A and B equals
the average effect of being assigned to treatment, the ITT.
The CACE is obtained by taking the ITT and dividing by
the proportion of Compliers in the sample
\[CACE = \frac{ITT}{Compliance Rate} =
\frac{2}{0.67} = 3 \]
- Intuitively, this operation is equivalent to redrawing the figure so
as to exclude Never-Takers. If Compliers ranged from 0 to 1 on the
X-axis, the shaded area in the treatment group would be 7, the shaded
area in the control group would be 4, and the difference would be
3.
Some subtle points:
- The figure illustrates why the “true” ATE cannot be estimated from
an experiment with non-compliance.
- The empty box in Panel A represents the average treatment effect
among Never-Takers.
- If Never-Takers were (somehow) exposed to the treatment, the empty
region and the cross-hatched region would together sum to the “true”
ATE.
- But the empty box never materializes because our experiment fails to
treat Never-Takers.
- Increasing the compliance rate could raise or lower
the CACE.
- Increasing the compliance rate means increasing the denominator in
the CACE equation (so we divide by a larger number)
- But in our specific example, treating never-takers would also
increase the numerator
- Relatedly, the “true” ATE is a weighted average of the ATE for
compliers and the ATE for never-takers.
- But, in the real world (unlike our hypothetical example here with
full info), you have no idea whether these two quantities are the
same.
- So be careful about generalizing the CACE to make inferences about
the ATE for the sample as a whole.
Another problem to work through
Suppose you conducted a field experiment where canvassers visited
homes and encouraged residents to recycle.
When implementing the intervention, you encountered one-sided
noncompliance: 1015 of the 1849 homes assigned to the treatment group
were successfully canvassed; none of the 1430 homes assigned to the
control group were canvassed.
When measuring outcomes three weeks later, you find that:
- 591 out of 1849 homes in the treatment group recycled
- 377 out of 1430 homes in the control group recycled
You also observed that:
- 429 out of 1015 homes that were successfully canvassed recycled
- 539 out of 2264 homes that were not canvassed recycled
Questions:
- What is the ITT?
- What is the CACE?
- Explain why comparing the recycling rates of the treated and
untreated subjects tends to produce misleading estimates.
Assumptions
Non-interference:
- Potential outcomes (Y_i) and treatment status (D_i) are
only affected by my own treatment assignment (Z_i).
- Neither Y_i nor D_i is affected by other subjects’
treatment assignment (Z_j).
Exclusion restriction:
- The only way that treatment assignment (Z) affects outcomes (Y) is
through the treatment itself (D), and not through some other
channel.
- By implication, treatment assignment should have no effect no
never-takers.
Here’s an example of a violation (e.g. if someone is not home when
canvassers visit, canvassers leave a flyer encouraging people to
vote):

Estimating using instrumental variables (IV) regression
Download a toy example of the recycling data here.
First stage: regress treatment status (D) on
treatment assignment (Z)
\[D = \alpha_1 + Z +
\epsilon_1\]
- The strength of this relationship = compliance rate = the strength
of your instrument
Second stage: regress outcomes (Y) on
predicted treatment status (\(\hat{D}\)) from the first stage
\[Y = \alpha_2 + \hat{D} +
\epsilon_2\]
- Notice that this regression invokes the exclusion restriction, since
Z is not part of the model
You can do both simultaneously using 2sls (ivregress
command in STATA).
With a small proportion of compliers,
- even a small violation of the exclusion restriction leads to severe
bias (since we are “scaling up” the ITT)
- your 2sls estimates of the CACE will very noisy
\[SE(\hat{CACE}) =
\frac{SE(\hat{ITT})}{ComplianceRate}\]